This paper presents a novel method for efficiently solving a trajectory planning problem for swarm robotics in cluttered environments. Recent research has demonstrated high success rates in real-time local trajectory planning for swarm robotics in cluttered environments, but optimizing trajectories for each robot is still computationally expensive, with a computational complexity from $O\left(k\left(n_t,\varepsilon \right)n_t^2\right)$ to $ O\left(k\left(n_t,\varepsilon \right)n_t^3\right)$ where $n_t$ is the number of parameters in the parameterized trajectory, $\varepsilon$ is precision and $k\left(n_t,\varepsilon \right)$ is the number of iterations with respect to $n_t$ and $\varepsilon$. Furthermore, the swarm is difficult to move as a group. To address this issue, we define and then construct the optimal virtual tube, which includes infinite optimal trajectories. Under certain conditions, any optimal trajectory in the optimal virtual tube can be expressed as a convex combination of a finite number of optimal trajectories, with a computational complexity of $O\left(n_t\right)$. Afterward, a hierarchical approach including a planning method of the optimal virtual tube with minimizing energy and distributed model predictive control is proposed. In simulations and experiments, the proposed approach is validated and its effectiveness over other methods is demonstrated through comparison.

翻译：本文提出一种在杂乱环境中高效求解集群机器人轨迹规划问题的新方法。近年来，已有研究证明了在杂乱环境中实现集群机器人实时局部轨迹规划的高成功概率，但每个机器人的轨迹优化仍面临较高的计算成本，其计算复杂度介于 $O\left(k\left(n_t,\varepsilon \right)n_t^2\right)$ 与 $O\left(k\left(n_t,\varepsilon \right)n_t^3\right)$ 之间，其中 $n_t$ 为参数化轨迹的参数数量，$\varepsilon$ 为精度，$k\left(n_t,\varepsilon \right)$ 为与 $n_t$ 和 $\varepsilon$ 相关的迭代次数。此外，集群难以实现群体协同运动。为解决此问题，我们定义并构建了包含无穷多条最优轨迹的最优虚拟管道。在特定条件下，最优虚拟管道中的任意最优轨迹均可表述为有限条最优轨迹的凸组合，其计算复杂度为 $O\left(n_t\right)$。随后，我们提出了一种分层方法，该方法融合了以能量最小化为目标的最优虚拟管道规划与分布式模型预测控制。通过仿真与实验验证了所提方法的有效性，并通过对比分析证明了其相对于其他方法的优越性。